Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933), страница 5
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Spherical coordinates for d’Alembert’s atmospheric tides. The fat linerepresents the visible part of the equator, over which the luminary is orbiting.N is the North pole.Suppose, with d’Alembert, that the tide-inducing luminaryorbits above the equator (with respect to the Earth).33 Usingthe modern terminology for spherical coordinates, call θ thecolatitude of a given point of the terrestrial sphere with respectto an axis pointing toward the orbiting luminary, φ the longitudemeasured from the meridian above which the luminary isorbiting (this is not the geographical longitude), η the elevationof the free surface of the fluid layer over its equilibriumposition, vθ and vφ the θ- and φ-components of the fluidvelocity with respect to the Earth, h the depth of the fluid inits undisturbed state, and R the radius of the Earth (see Fig.
6).D’Alembert first considered the simpler case when φ isnegligibly small, for which he expected the component vφ alsoto be negligible. To first order in η and v, the conservation ofthe volume of a vertical column of fluid yields:11 ∂vθvθη̇ ++= 0,(15)hR ∂θR tan θwhich means that an increase of the height of the column iscompensated for by a narrowing of its basis (the dot denotes thetime derivative at a fixed point of the Earth surface). Since thetidal force f is much smaller than the gravity g, the vector sumf + g − γ makes an angle ( f θ − γθ )/g with the vertical.
To firstorder in η, the inclination of the fluid surface over the horizontalis (∂η/∂θ )/R. Therefore, the condition that f + g − γ shouldbe perpendicular to the surface of the fluid is approximatelyidentical to34g ∂η.(16)R ∂θAs d’Alembert noted, this equation of motion can also beobtained by equating the horizontal acceleration of a fluid sliceγθ = f θ −33 The sun and the moon actually do not, but the variable part of their actionis proportional to that of such a luminary.34 D’Alembert, 1747: 88–89 (formulas A and B). The correspondence withd’Alembert’s notation is given by: θ 7→ u, vθ 7→ q, ∂η/∂θ 7→ −v, R/ hω 7→ ε,R/gK 7→ 3S/4 pd 3 (with f = −K sin 2θ).O.
Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869to the sum of the tidal component f θ and of the differencebetween the pressures on both sides of this slice. Indeed, theneglect of the vertical acceleration implies that at a given height,the internal pressure of the fluid varies as the product gη.Hence, d’Alembert was aware of two routes to the equationof motion, through his dynamic principle, or through anapplication of the momentum law to a fluid element subjectedto the pressure of contiguous elements.
In some sections hefavored the first route, in others the second.35In his expression of the time variations η̇ and v̇θ , d’Alembertconsidered only the forced motion of the fluid for which thevelocity field and the free surface of the fluid rotate togetherwith the tide-inducing luminary at the angular velocity −ω.Then the values of η and vθ at the colatitude θ and at the timet + dt are equal to their values at the colatitude θ + ωdt and atthe time t.
This givesv̇θ = ω∂vθ,∂θη̇ = ω∂η.∂θ(17)D’Alembert equated the relative acceleration v̇θ with theacceleration γθ , for he neglected the second-order convectiveterms, and judged the absolute rotation of the Earth asirrelevant (he was aware of the centripetal acceleration, buttreated the resulting permanent deformation of the fluid surfaceseparately; and he overlooked the Coriolis acceleration). Withthese substitutions, his Eqs. (15) and (16) become ordinarydifferential equations with respect to the variable θ.D’Alembert eliminated η from these two equations, andintegrated the resulting differential equation for Newton’s value−K sin 2θ of the tide-inducing force f θ .
In particular, heshowed that the phase of the tides (concordance or opposition)depended on whether the rotation period 2π/ω √of the luminarywas smaller or larger than the quantity 2π R/ gh, which hehad earlier shown to be identical with the period of the freeoscillations of the fluid layer.36In another section of his memoir, d’Alembert extended hisequations to the case when the angle φ is no longer negligible.Again, he had the velocity field and the free surface of thefluid rotate together with the luminary at the angular velocity−ω. Calling Rωdt the operator for the rotation of the angle ωdtaround the axis joining the center of the Earth and the luminaryand v(P, t) the velocity vector at point P and at time t, we have:∂vθ∂vθ sin φcos φ −− vφ sin φ sin θ ,∂θ∂φ tan θ∂vφ∂vφ sin φv̇φ = ωcos φ −+ vθ sin φ sin θ .∂θ∂φ tan θv̇θ = ω1863(19)(20)For the same reasons as before, d’Alembert identified thesederivatives with the accelerations γθ and γφ .
He then appliedhis dynamic principle to get:g ∂η,R ∂θg ∂η.γφ = −R sin θ ∂φγθ = f θ −Lastly, he obtained the continuity condition:∂η∂η sin φη̇ = ωcos φ −∂θ∂φ tan θ∂vθvθ1 ∂vφ++=−,∂θtan θsin θ ∂φ(21)(22)(23)in which the modern reader recognizes the expression of adivergence in spherical coordinates.37D’Alembert judged the resolution of this system to bebeyond his capability. The purpose of this section of his memoirwas to illustrate the power and generality of his method forderiving hydrodynamic equations. For the first time, he gavethe complete equations of motion of an incompressible fluid ina genuinely two-dimensional case. Thus emerged the velocityfield and partial derivatives with respect to two independentspatial coordinates.
Although Alexis Fontaine and Euler hadearlier developed the needed calculus of differential forms,d’Alembert was first to apply it to the dynamics of continuousmedia. His notation of course differed from the modern one:where we now write ∂ f /∂ x, Fontaine wrote d f /dx, andd’Alembert often wrote A, with d f = Adx + Bdy + · · · .3.4. The resistance of fluidsExpressing this relation in spherical coordinates, d’Alembertobtained:In 1749 d’Alembert submitted a Latin manuscript on theresistance of fluids for another Berlin prize, and failed to win.The Academy judged that none of the competitors had reachedthe point of comparing his theoretical results with experiments.D’Alembert did not deny the importance of this comparisonfor the improvement of ship design.
But he judged that therelevant equations could not be solved in the near future, andthat his memoir deserved consideration for its methodologicalinnovations. In 1752, he published an augmented translation ofthis memoir as a book.3835 D’Alembert, 1747: 88–89. He represented the internal pressure by theweight of a vertical column of fluid. In his discussion of the conditionof equilibrium (1747: 15–16), he introduced the balance of the horizontalcomponent of the external force acting on a fluid element and the differenceof weight of the two adjacent columns as “another very easy method” fordetermining the equilibrium. In the case of tidal motion with φ ≈ 0, he directlyapplied this condition of equilibrium to the “destroyed motion” f + g − γ .
Inthe general case (D’Alembert, 1747: 112–113), he used the perpendicularity off + g − γ to the free surface of the fluid.36 The elimination of η leads to the easily integrable equation(gh − R 2 ω2 )dvθ + ghd(sin θ )/ sin θ − R 2 ωK sin θd(sin θ ) = 0.37 D’Alembert, 1747: 111–114 (Eqs. E, F, G, H, I). To complete thecorrespondence given in note (36), take φ 7→ A, vφ 7→ η, γθ 7→ π , γφ 7→ ϕ,g/R 7→ p, ∂η/∂θ 7→ −ρ, ∂η/∂φ 7→ −σ , ∂vθ /∂θ 7→ r , ∂vθ /∂φ 7→ λ,∂vφ /∂θ 7→ γ , ∂vφ /∂φ 7→ β.
D’Alembert has the ratio of two sines insteadof the product in the last term of Eqs. (19) and (20). An easy, modern wayto obtain these equations is to rewrite (18) as v̇ = [(ω × r) · ∇]v + ω × v,with v = (0, vθ , vφ ), r = (R, 0, 0), ω = ω(sin θ sin φ, cos θ sin φ, cos φ), and∇ = (∂r , ∂θ /R, ∂φ /(R sin θ )) in the local basis.38 D’Alembert, 1752: xxxviii.
For an insightful study of d’Alembert’s workon fluid resistance, cf. Grimberg, 1998 (which also contains a transcript ofthe Latin manuscript submitted for the Berlin prize). See also Calero, 1996:Chapter 8.v(P, t + dt) = Rωdt v(Rωdt P, t).(18)1864O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869Compared with the earlier treatise on the equilibrium andmotion of fluids, the first important difference was a newformulation of the laws of hydrostatics. In 1744, d’Alembertstarted with the uniform and isotropic transmissibility ofpressure by any fluid (from one part of its surface to another).He then derived the standard laws of this science, such asthe horizontality of the free surface and the depth-dependenceof wall-pressure, by qualitative or geometrical reasoning.
Incontrast, in his new memoir he relied on a mathematicalprinciple borrowed from Alexis-Claude Clairaut’s memoir of1743 on the shape of the Earth. According to this principle, afluid mass subjected to Ra force density f is in equilibrium ifand only if the integral f · dl vanishes over any closed loopwithin the fluid and over any path whose ends belong to thefree surface of the fluid.39D’Alembert regarded this principle as a mathematical expression of his earlier principle of the uniform transmissibilityof pressure. If the fluid is globally in equilibrium, he reasoned,it must also be in equilibrium within any narrow canal of section ε belonging to the fluid mass. For a canal beginning andending on the free surface of the fluid, the pressure exerted bythe fluid on each of the extremities of the canal must vanish. According to the principle of uniform transmissibility of pressure,the force f acting on the fluid within the length dl of the canalexerts a pressure ε f · dl that is transmitted to both ends of thecanal (with opposite signs).RAs the sum of these pressures mustvanish, so does the integral f·dl.
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